Application of isotonic regression in estimating ED _g and its 95% confidence interval by bootstrap method for a biased coin up-and-down sequential dose-finding design

Abstract

INTRODUCTION Newer anaesthetic drugs introduced in practice and older drugs for novel route or indication has led to the need to determine the optimal dose. In this context, two most recent studies, for example, estimated the effective dose of opioids for epidural initiation in the latent and active phases during the first stage of labour and remimazolam bolus for anaesthesia induction in different age groups.[1 2] The primary aim of these studies was to estimate a quantile, a dose at which a desired probability of response is achieved. This dose is called an effective dose (ED) with quantile g; that is, EDg is defined as the dose of a drug that produces a response of interest at quantile ‘g’ (‘g’ may be 90, 95 or 99%) of the study population. In these studies, doses cannot be assigned randomly because some patients may receive optimal low doses, whereas others may receive high doses that might induce adverse effects. To address this issue, scholars demonstrated the up-and-down design (UDM) to estimate the median, whereas Derman demonstrated through nonparametric experimentation that dose levels could be centred around any given target quantile using the up-and-down designs using a biased coin.[3 4] Later, Durham et al.[5] generalised UDM as a biased coin design to estimate EDg using random walk rules. This method of the biased coin up-and-down design (BCUD) is used to assign doses sequentially by random walk rule in which efficacy or toxicity is assumed to be monotonically related to dose. In dose-finding studies within the BCUD setting, the isotonic regression technique is utilised to estimate the effective dose for a particular drug (EDg). Unlike simple linear regression, isotonic regression ensures that the regression function is monotonic, meaning it continuously increases or decreases. This method is suitable because it assumes that increasing the dose level increases the drug effect.[6] In this regression, the dose (x) serves as the predictor variable, whereas the response probability (P (x) = P (response|dose x)) acts as the dependent variable, representing the probability of a response to the drug. This paper aims to describe these mathematical concepts using a diagrammatic explanation, specifically focusing on a section of the isotonic regression curve [Figure 1]. The paper also covers the estimation of naïve probability and adjusted probability using the pooled adjacent violators algorithm (PAVA) for the response. Figure 1 A small section of isotonic regression curve The data used to elucidate the calculations needed to estimate ED90 and its 95% confidence interval (CI) are given [Table 1 and Figure 2a] in annexure. Table 1 Norepinephrine prophylactic bolus dose and the response of 40 successive women Patient number Norepinephrine Prophylactic bolus dose (μg) Response# Patient number Norepinephrine Prophylactic bolus dose (μg) Response# 1 4 F 21 11 S 2 5 F 22 11 F 3 6 F 23 12 S 4 7 S 24 12 S 5 7 S 25 12 S 6 7 S 26 12 S 7 7 S 27 12 S 8 7 S 28 12 S 9 7 F 29 12 S 10 8 S 30 12 S 11 8 S 31 11 S 12 8 F 32 11 S 13 9 S 33 11 S 14 9 S 34 11 S 15 9 S 35 11 S 16 9 F 36 11 S 17 10 F 37 11 S 18 11 S 38 11 S 19 11 S 39 11 S 20 11 S 40 11 S # F is failure, and S is success Figure 2 Plots showing the (a) patient’s allotment sequence and the response to the assigned dose of norepinephrine prophylactic bolus (μg) and (b) naïve probability (observed response rate) and PAVA probability (adjusted response rate) for the assigned dose Estimation of naive and PAVA probability Table 2 shows the dose assigned to the participants (nDoses), number of patients (nTrials), number of successes (nEvents), the naïve probability and PAVA adjusted response rate at each distinctive dose level. Table 2 Naive (observed) and PAVA probability and bootstrap estimates of 3000 boot replications nDoses nTrials nSuccess Naive probability PAVA probability 4 1 0 0 0 5 1 0 0 0 6 1 0 0 0 7 6 5 0.8333 0.7143 8 3 2 = 0.6667 0.7143 9 4 3 = 0.75 0.7143 =10 1 0 0 f ()=0.7143 =11 15 14 0.9333 f ()=0.9333 12 8 8 1 1 Statistic from bootstrapping (3000 replications) Value Original statistic (ED90) 10.848 Mean of ED90’s of Boot Replications 10.772 Median of ED90’s of Boot Replications 10.834 Bias of Original statistic (ED90) (= Original Statistics – Mean of ED90’s of Boot Replications) -0.076 Bias Correction: (No. of Boot Replicates <=Original Statistic)/(Total Replicates +1) 0.51583 Standard error of Boot Statistic (Mean of ED90’s) 0.626 0.03969 zα/2 -1.96 z(1-α/2) 1.96 2.5% Bias-corrected Lower Bound 9.25 97.5% Bias-corrected Upper Bound 11.675 The naive probability is calculated [Table 2] using the following formula: The PAVA probability is estimated using the PAVA algorithm for each dose level.[6] Under the PAVA algorithm, starting with the lowermost dose, we have to find the first adjoining pair of naive probability that violates the increasing ordering restriction (i.e. increasing dose level increases the drug effect), that is where . The PAVA probability for that pair of doses is In Figure 2b, the first adjoining pair of doses that violate the ordering restriction of naive probability is 7 and 8 μg, and the next successive pair of such doses is 9 and 10 μg, but for dose 10 μg, the naive probability is zero. Hence, the PAVA probability using equation (B) for doses 8 and 9 μg is The same PAVA probability of 0.7143 is taken for preceding and succeeding doses of 7 and 10 μg [Table 2]. ED90 and its 95% confidence interval By substituting the values from Table 2 in annexure equation (8), the ED90 is given below: The precision of ED90 is its 95% CI and is estimated using annexure equations (11) and (12): The final estimation of 95% bias-corrected bootstrap confidence interval (BCBCI) is estimated using the annexure equation (13): Hence, the 95% CI for ED90 is (9.25, 11.675) μg. All calculations were performed using R 4.2.1 (R Foundation of Statistical Computing, Vienna, Austria), and the code used is given in Appendix I. The bias of the original statistic (ED90) = -0.076 is basically due to the discrete nature of the doses rather than the dimensional. The bias would be closer to zero if the dose ranges continuously, say, 10.1, 10.2, 10.3,…, 11.0, which would make the distance closer between the PAVA probability 0.7143 and 0.9000 for the dose of 10 μg. DISCUSSION We explored the application of isotonic regression for estimating EDg along with its 95% confidence interval in the context of dose-finding studies. In anaesthesia, it is vital to assess how drug effects change with increasing doses using dose–response characterisation. In a general scenario, we may encounter a violation of the assumption of monotonicity in the observed probability, as occurred in the specified example. We employed the PAVA algorithm to rectify this violation and ensure the validity of the assumption. The isotonic estimate of ED90 in the BCUD setting has low bias and variance, particularly at low or high quantiles.[7] It also has a smaller mean square error than estimators from other methods.[8 9 10] We also assessed the overfitting of the estimate using bootstrapping, and the bias of the ED90 estimate was very low [-0.076, Table 2]. To evaluate the precision of the estimated target dose, several methods estimate the bootstrap confidence interval with a confidence level of (1-α)*100%.[11] Simulation studies suggested that the BCBCI method, described in the paper, provides better balance, increased type I error and higher power than other bootstrap methods.[11 12] Sequential dose-finding studies are appealing because they provide accurate and stable estimates with small sample sizes ranging from 20 to 40 patients.[5 8] These studies determine the critical intensity level (dose) at which a drug either produces or prevents a reaction in each patient. In such studies, the dose increment between the first dose at experimentation and the following subsequent doses is very small, and therefore, the outliers are unlikely. The comparative analysis of the proposed technique against other methods is not included in this paper as it is beyond the scope of the paper and can be found elsewhere.[8] However, one notable comparison is the simplicity of the BCUD method compared to other methods; for example, the continual reassessment method requires a mathematical model to assign the dose and analyse previous dose responses for the next dose, which requires the involvement of a biostatistician.[13] On the other hand, the BCUD sequential method requires less computation and does not rely on a mathematical model to assign the doses. It also has simple statistical properties for estimating the target EDg.[6] In contrast, other standard methods like logit and probit regression are complex and hence not widely used by anaesthesiologists. The isotonic estimate of EDg, on the other hand, does not demand technical expertise, and that is why anaesthesiologists should prefer this method to estimate any target dose without the help of the biostatistician. CONCLUSION The isotonic regression adjusts the observed response rate (naive) using PAVA when it is not monotonically increasing with increasing dose levels. Also, it is straightforward to estimate the effective dose for a target quantile in the BCUD setting. Additionally, computer programming is needed only to estimate the 95% CI of EDg. Financial support and sponsorship Nil. Conflicts of interest There are no conflicts of interest.